24 May 2021 Edvard@Fysik.Su.Se Orsodn Uhr Dadm¨Ortsell Edvard Author: Corresponding Tension

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ABSTRACT 1 σ 1 olJohansson, Joel H easi h DOM apeo ee esdquasars lensed seven yield of sample TDCOSMO the in delays 2017 observations (TRGB) branch giant us- red gives the magnitude of SNIa tip the absolute ing the Calibrating nificant: eto-trsse oae oteglx G 4993 NGC galaxy binary the a to of at located merger gravita- system the the neutron-star GW170817, of signal amplitude wave the tional from derived is Universe in- CMB the and SNIa values. calibrated ferred Cepheid the tween T07f,yedn ubecntn ihrelatively of with uncertainties constant large Hubble a yielding AT2017gfo, 0 otepeiinrqie ocamasubstantial a claim to required precision the to ieetrueivle rvttoa esn.Time lensing. gravitational involves route different A h eso ewe te esrmnsi o ssig- as not is measurements other between tension The nte oa siaeo h xaso aeo the of rate expansion the of estimate local Another z ). 0 = H H 0 0 . 69 = 1 hog h lcrmgei counterpart, electromagnetic the through 010 74 = ni oos yielding colors, insic c fCpedclrlmnst cali- color-luminosity Cepheid of ice itosi nrni oos resulting colors, intrinsic in riations 1 -aatcetnto yds,w re- we dust, by extinction a-galactic oo-uioiyrlto ocorrect to relation color-luminosity ,acmo rciei h field the in practice common a s, n ualDhawan Suhail and . nmto,weetemeasured the where method, en . 6 5 beconstant, bble rec aayidvdal.We individually. galaxy each or ± − +5 6 1 . . 6 1 . oo Calibration Color d ( 6 ( rsity irre l 2020 al. et Birrer remne l 2019 al. et Freedman H a 12) Interstellar (1728), vae 0 70 = H H 0 rmTp Ia Type from , . 0 nd. 2 0 71 = − +12 8 . 0 ah we oach, . 0 .Constraining ). . 8 ( ± bote al. et Abbott ,rgtbe- right ), 1 . 6 2 Mortsell¨ et al. the galaxy lens mass profiles, using kinematics obser- present before CMB photon decoupling, e.g., new ther- vations of an independent set of gravitational lenses in mal relativistic species or early dark energy, or by reduc- the Sloan Lens ACS sample (SLACS), lowers the value ing the sound speed. However, such modifications are +4.1 to H0 = 67.4−3.2, assuming that the TDCOSMO and severely constrained when taking the full CMB power SLACS galaxies are drawn from the same parent pop- spectrum into account. Attempts to shift the CMB ulation. These results are illustrated in Figure 1, from value also involve changing the expansion history at red- which is evident that only the Cepheid calibrated SNIa shifts z < 1090, with modest success since the expansion distance scale from SH0ES is in definite tension with the rate is tightly constrained by SNIa and baryonic acoustic CMB inferred distance. oscillation (BAO) observations. Given that the proposals mentioned above require substantial modifications of the current concordance cosmological model, and still fail in relieving the full 1000 CMB tension, we investigate the Cepheid-SNIa value and its 100 Lensing uncertainties, see also Follin & Knox (2018); Efstathiou SH0ES (2020). In particular, we concentrate on dust extinction, TRGB-SNIa 10 GW affecting all astronomical observations in the optical and near infrared (NIR) regime. We focus on a a very spe- z 1 cific assumption made by the SH0ES team throughout their series of publications, namely that there is a uni- 0.1 versal reddening law in all galaxies.

0.01 2. REVISITING DUST EXTINCTION CORRECTIONS 60 65 70 75 80 85 H In spite of the critical importance for precision cosmol- 0 ogy, the current understanding of light attenuation in the interstellar medium (ISM) of galaxies remains very Figure 1. The Hubble constant as inferred from distances measured to the CMB, strongly lensed quasars, SNIa cal- limited. In comparison, the Milky Way (MW) ISM has ibrated with Cepheids (SH0ES) and the TRGB, and the been studied in great detail, including the properties of gravitational wave signal GW170817. The redshift corre- dust grains responsible for dimming of light(see Draine sponds to the mean redshift of the distances employed in the 2003, for a review). In particular, several MW redden- method, slightly shifted where needed to avoid overlap. The ing laws have been devised, among these Cardelli et al. major tension is between the values inferred from Cepheid (1989) (CCM), O’Donnell (1994) and Fitzpatrick (1999) calibrated SNIa by the SH0ES team and CMB observations. (F99). They have in common the use of a single param- BV eter, the total to selective extinction coefficient, RV , as a proxy for the grain composition and size distri- The value inferred from the CMB depends on the en- bution, where the attenuation in the optical V-band tire expansion history of the Universe, whereas the SNIa relates to the color excess E(B − V) ≡ AB − AV as BV measurement only depends on the local expansion rate AV = RV E(B − V), here referred to as the ”CCM- BV it sets out to measure. On the other hand, the infer- relationship”. Low values of RV indicate a steep wave- BV ence from SNe Ia depends on a combination of a larger length dependence, while large RV correspond to gray number of astrophysical probes. Therefore, attempts to extinction. BV modify the CMB inferred H0 usually rely on modifica- While an average hRV i =3.1 for the MW is found in tions of the cosmological model, whereas the SNIa value most studies, significant variations are found in individ- BV is usually studied with emphasis on possible systematic ual lines of sight in the galaxy, ranging from RV ≈ 2 in BV effects concerning the local distance measurements. some diffuse sight lines, to RV ≈ 6 in dense molec- At least in principle, the CMB value can be increased ular clouds (Fitzpatrick 1999). The diversity in the by various departures from the concordance cosmolog- MW has been confirmed by Nataf et al. (2016), who BV ical constant and cold dark matter, ΛCDM model, see find significantly lower values of RV in the Galactic e.g. Mörtsell & Dhawan (2018); Knox & Millea (2020). bulge. Moving the scope outside the MW, a study Options include decreasing the physical size of the sound by Gordon et al. (2003) of the extinction in the Mag- horizon used to measure the distance to z = 1090. ellanic Clouds found that a small number of LMC ex- This can be accomplished by adding sources of energy tinction curves are consistent with the CCM relation- The Hubble Tension Bites the Dust 3 ship, but the majority of the LMC and all the SMC curves are not. Fausnaugh et al. (2015) report a gray 1.0 extinction law for NGC 4258 in the line of sight of BV the Cepheids, RV ∼ 4.9, although they caution that 0.8 this could be the result of unresolved systematics. For more distant galaxies, observed SNIa colors highlight 0.6

the observed diversity in extinction properties, rang- VI H ing from RBV ∼ 1 to values consistent with the MW R V 0.4 FTZ19 average (see Krisciunas et al. 2006; Nobili & Goobar B18 SH17 2008; Goobar et al. 2014; Amanullah et al. 2014, 2015; LMC1 Burns et al. 2018, and references therein). For the 0.2 LMC2 FA15 SNe Ia in the Hubble flow, color corrections are based 0.0 on the SALT2 lightcurve fitter (Guy et al. 2007), which 1 2 3 4 5 6 again differ from the CCM parameterization, but are RBV BV V most consistent with values of RV ∼ 2.5 (see e.g., Biswas et al. 2021, and references therein), although VI BV dust extinction differences between host galaxy envi- Figure 2. RH vs RV relation and 1σ scatter from the extinction law derived by Fitzpatrick et al. (2019) (FTZ19), ronments has been suggested as an explanation for BV BV extrapolated to RV > 6 and RV < 2. Such low val- a systematic ”mass step” in the derived distances ues have been inferred from SNIa colors (Burns et al. 2018, (Brout & Scolnic 2021; Johansson et al. 2021). BV B18). Also shown are derived RV values from a MW stel- To minimize the impact from extinction correction lar sample in Schlafly et al. (2017) (SH17), two samples in uncertainties, the SH0ES team use flux measurements the LMC (Gordon et al. 2003, LMC1 and LMC2) and in the in the NIR H-band, centered at 1.6 µm, where extinc- water mega-maser anchor NGC 4258 (Fausnaugh et al. 2015, tion by dust, gauged using the observed color V − I, FA15). is significantly smaller. Adopting the CCM-like relationship from F99 and the extinction correction AH = ter RW is argued to also partly correct for an intrinsic VI RH E(V − I), the value corresponding to the MW aver- Cepheid C-L relation. VI age is RH ∼ 0.4. However, there is no theoretical, nor In Follin & Knox (2018), a slightly different C-L cor- any empirical studies of extinction suggesting that a uni- rection with respect to an estimated color excess of VI ˆ ˆ versal value of RH can be assumed. On the contrary, a RE E (V − I) was employed, where E (V − I) ≡ (V − recent study by Fitzpatrick et al. (2019) finds consider- I) − hV − Ii0 with hV − Ii0 an estimate of the mean able variations between lines of sight for the extinction intrinsic Cepheid color. Here, RE is interpreted as the VI curves in the NIR in the Milky-Way. Based on a pa- dust total to selective extinction ratio, RH . Imposing a rameterization fitting extinction laws from ultraviolet to prior of RE =0.39 ± 0.1, a value of H0 = 73.3 ± 1.7 was VI NIR for 72 well-measured stars, a very wide range in RH derived when allowing RE to vary between galaxies, in can be inferred, as shown in Figure 2. Hence, assuming good agreement with H0 = 73.2 ± 1.3 from Riess et al. VI a narrow range in RH for the anchor and Cepheid hosts (2021). At face value, this result seems to suggest that in not warranted by current observations. In this paper, the method used for calibrating the C-L relation and/or we investigate to what degree relaxing this assumption the possibility of varying this calibration between galax- affects the inferred value of H0 and its corresponding ies have a small impact on the inferred Hubble constant. uncertainties. As argued below, regardless of whether calibrating with respect to the observed color or an estimated color 3. DIFFERENCE IN METHODOLOGY excess, the correction should apply for both dust extinction and the intrinsic Cepheid C-L relation. Since both The local distance ladder uses the difference between contributions are uncertain at NIR wavelengths, and at Cepheid magnitudes in SNIa hosts and anchor galaxies. least dust extinction properties are known to vary be- Therefore, it is relatively insensitive to changes in the tween galaxies, we investigate the effect of allowing for global properties of the extinction law. In Riess et al. R and R to do the same. Given the lack of solid in- (2016), a global change of the parameter R parameter- W E W dependent constraints on R , we employ less restrictive izing a Cepheid color-luminosity (C-L) correction with E priors than Follin & Knox (2018). Any systematic dif- respect to the observed color R (V−I) of R =0.39 → W W ference in dust properties or the intrinsic C-L between 0.35 changed the Hubble constant by δH ∼ 0.5. When 0 SNIa hosts and anchor galaxies will shift the inferred multiplied with the observed color V − I, the parame- 4 Mortsell¨ et al. value of H0. We will show that such a difference is sup- and a lower limit to the precision in H0 is set by the ported by the Cepheid and SNIa data. precision of the anchor distance measurements. Shifting δ(∆mCeph) = ±0.1 will shift the Hubble constant by 4. METHOD AND DATA δH0 ∓ 4.6 %. The apparent magnitude of a source at redshift z with absolute magnitude M is given by 4.1. Cepheid calibration We use the Hubble Space Telescope (HST) flux in the m = 5 log D(z)+ M + 25, (1) NIR filter (H = F160W) band, color calibrated using optical (V = F555W and I = F814W) data, to derive where D(z) is the luminosity distance in units of Mpc. Wesenheit magnitudes By combining observed magnitudes of Cepheids in SNIa host and anchor galaxies, mhost and manch with SNIa W Ceph Ceph mH ≡ mH−RW (V−I) = mH−RW E(V−I)−RW (V−I)0, magnitudes in host galaxies and in the Hubble flow, (6) mhost and mflow , we can derive SN SN where the color excess E (V − I) ≡ AV − AI = (V − I) − (V − I) , with (V − I) the intrinsic Cepheid color. In 5 log H = 5 log r(z) − 5 log Danch + ∆m − ∆m , 0 0 0 SN Ceph the last step, we see that mW is corrected both for dust (2) H extinction, identifying the first RW with the total to se- where r(z) ≡ H0D(z) can be approximated by r(z) ≈ cz VI lective extinction ratio R ≡ AH/(AV − AI), and for in the close Hubble flow and we have defined H a possible intrinsic C-L relation, identifying the second VI host flow RW with β , as parameterized in e.g. Madore (1982) ∆mSN ≡ m − m , (3) H SN SN and Madore et al. (2017). The term βVI (V − I) cor- host anch H 0 ∆mCeph ≡ mCeph − mCeph. (4) responds to the intrinsic magnitude-color relation at a fixed Cepheid period, whereas the correlation between Apart from getting the SNIa redshifts and anchor dis- the intrinsic color with period is included in the period- tances right, we thus need to make sure there are no luminosity (P-L) calibration parameterized by b in systematic offsets in the Cepheid and SNIa magnitudes W equation 9 below. As an alternative approach, also em- between host, anchor and cosmic flow galaxies. Ignoring ployed in Follin & Knox (2018), we calibrate the C-L the weak cosmology dependence of r(z) (Dhawan et al. relation using 2020), the inferred value of H0 will decrease (increase) if we: W ˆ mH ≡ mH − RE E (V − I). (7) 1. Increase (decrease) the independent anchor dis- Here, Eˆ (V − I) represents a proxy for the color excess tances, Danch. obtained by subtracting an estimate of the mean intrin- 1 2. Decrease (increase) ∆mSN. sic colors, hV − Ii0 from the observed colors ,

3. Increase (decrease) ∆mCeph. Eˆ (V − I) ≡ (V − I) − hV − Ii0. (8)

Here, we focus on option 3. With regards to option 2, The estimated color excess Eˆ (V − I) also represents a ∆mSN will increase if SNIa in Cepheid host galaxies are combination of dust extinction and intrinsic color, since systematically made brighter than in the Hubble flow, the mean intrinsic colors, hV − Ii0 does not take into e.g., if there is additional dust extinction not accounted account individual variations in Cepheid temperature for in the host galaxies, or if the effect that SNIa in high along the width of the Cepheid instability strip, see e.g. mass hosts are systematically brighter than in low mass Pejcha & Kochanek (2012). galaxies, such as Cepheid hosts, have been underesti- Using multi wavelength data, one can in principle at- mated (see Rigault et al. 2020, and references therein). tempt to distinguish the contribution from dust and In terms of option 3, if there is additional dust ex- intrinsic color variations, see e.g. Pejcha & Kochanek tinction not accounted for in the anchor galaxies, or if (2012); Madore et al. (2017) and calibrate them sepa- we have over-corrected for dust extinction in the host rately. In the following, we will follow standard practice galaxies, the inferred value of H0 will decrease, and vice and assume that RW and RE effectively corrects both for versa. The fractional shift in the Hubble constant is

anch 1 δH0 δr(z) δD ln 10 Note that if hV−Ii0 is assumed to depend on the Cepheid period, = − + [δ(∆mSN) − δ(∆mCeph)] , anch the fitted W parameterizing the P-L relation will shift accord- H0 r(z) D 5 b (5) ingly. The Hubble Tension Bites the Dust 5 dust and intrinsic color variations, but remain agnostic effectively transforming zp into a linear parameter, and whether this is most effectively done using the Wesen- 5 heit calibration using observed colors or with respect to mW − 10+ ln π = mW + b [P] H,j ln 10 H W j estimated color excesses. Taking an empirical approach, 5 zp in both cases, we will fit for the values of RW and RE + ZW[M/H]j − . (15) W ln 10 π that minimize the scatter in mH . We model the Wesenheit magnitude of the jth Higher order terms, O(zp/π)2, are small and corrected Cepheid in the ith SNIa host as for in an iterative manner. W W Type Ia Supernovae mH,i,j = µi + MH + bW[P]i,j + ZW[M/H]i,j , (9) 4.3. The calibrated SNIa B-band peak magnitude in the where [M/H] is a measure of the metallicity of the i,j ith host is modelled by Cepheid, [P]i,j ≡ log Pi,j − 1 where Pi,j is the period W measured in days, MH the absolute Cepheid magnitude mB,i = µi + MB. (16) normalized to a period of P = 10 days and Solar metallicity and µi the distance modulus to the ith galaxy. In The SNIa peak apparent magnitudes need to be cor- what follows, we will allow for separate P-L relations for rected for the width-luminosity and C-L relations. short and long period Cepheids using There are several lightcurve fitting algorithms for de- riving the SNIa peak magnitude, lightcurve shape and s s l l bW[P]i,j → bW[P]i,j + bW[P]i,j , (10) color, the most widely used for cosmology being the

s SALT2 model (Guy et al. 2007). The derived lightcurve where [P]i,j = 0 for Cepheids with periods > 10 days l widths and colors are used to correct the peak magni- and [P]i,j = 0 for Cepheids with periods < 10 days, see tude mB in equation 16. The errors on the corrected Section A. peak magnitude include the ﬁtting error and a 0.1 mag Similarly for the jth Cepheid in the kth anchor galaxy, term from the SNIa model added in quadrature. here MW, NGC 4258 and the Large Magellanic Cloud (LMC), 4.4. Data

W W For the extra-galactic (M31 and beyond) Cepheids, mH,k,j = µk + mH + bW[P]k,j + ZW[M/H]k,j . (11) including Cepheids in the anchor galaxy NGC 4258, we 4.2. Milky Way Cepheids use the data set from Table 4 in Riess et al. (2016). This Trigonometric parallaxes potentially provide the most table is restricted to Cepheids passing a best-fit, global direct calibration of the Cepheid absolute magnitude, 2.7 σ outlier rejection, the impact of which is claimed W to be small in Riess et al. (2016), but not possible to MH . We use data from Riess et al. (2021), with 75 MW Cepheids, out of which 68 have reliable GAIA par- confirm independently by us. allaxes. For the jth Cepheid in the MW, For Cepheids in the LMC, we use data in Table 2 in Riess et al. (2019). Data for MW Cepheids, includ- W W mH,j = µj + MH + bW[P]j + ZW[M/H]j . (12) ing GAIA parallax measurements are from Table 1 in Riess et al. (2021). where the distance modulii for each Cepheid is estimated Double eclipsing binaries (DEBs) provide a means to using GAIA parallaxes, π, according to measure distances by determining the physical sizes of the member stars via their radial velocities and light −0.2(µj −10) πj + zp = 10 , (13) curves (Paczynski 1996). 20 DEBs observed using long- where zp is a residual parallax calibration offset that we baseline near-infrared interferometry give a distance to W the LMC of µLMC = 18.477 ± 0.0263 (Pietrzyński et al. fit for together with MH ,bW and ZW. In Riess et al. W 2019; Riess et al. 2019). We use the updated distance to (2021), MH and zp are fit for using only MW data NGC 4258 of µLMC = 29.397 ± 0.032 (Reid et al. 2019), setting bW = −3.26 and ZW = −0.17 (as fitted to all Cepheids), finding zp = −14 ± 6 µas. Since we want to using observations of mega-masers in Keplerian motion fit for zp simultaneously with all parameter, we write around its central super massive black hole. Type Ia SN B-band magnitudes are from Table 5 in 5 zp Riess et al. (2016), derived using version 2.4 of SALT II µj = 10 − ln π + ln 1+ ln 10 h π i (Betoule et al. 2014). 5 zp zp 2 = 10 − ln π + + O , (14) 4.5. Parameter Fitting ln 10 π π 6 Mortsell¨ et al.

Given the observed Cepheid magnitudes mH, colors down to the Planck value. This could be argued for, e.g., V−I, periods [P], metallicities [M/H], together with the if dust properties in SNIa host galaxies have a system- SNIa magnitudes mB, the anchor distances µk and the atically steeper extinction law than the anchor galaxies. MW Cepheid parallaxes π, we can fit simultaneously for In lack of solid independent evidence for such a system- RW or RE, bW, ZW, the host galaxy distances µi, the atic shift, or the value of RW and possible variations of anchor distances µk, the GAIA parallax offset zp, the it, we will next include RW as a model parameter to be W Cepheid absolute magnitude MH and the SNIa abso- constrained by the available data. lute magnitude MB. Since the system of equations is linear, the fit can be made analytically as described in Section A. 0.6 88 Given MB, the Hubble constant is calculated as 68 62 84 MB/5+aB+5 H0 = 10 (17) 70 0.5 64 81 where aB is the intercept of the SNIa magnitude-redshift 72 relation 66 77 74 aB+mB/5 1 78 10 = cz 1+ [1 − q0] z 0.4 76 74 2 68 (anchors)

1 W − 1 − q − 3q2 + j z2 + O(z3) (18) R 70 6 0 0 0 70 0.3 67 measured with q0 = −0.55 and j0 =1to aB =0.71273± 72

0.00176 (Riess et al. 2016). 63 74 76 84 78 80 82 86 5. RESULTS FOR THE WESENHEIT 0.2 60 CALIBRATION RW (V − I) 0.2 0.3 0.4 0.5 0.6 R (hosts) In Riess et al. (2021), a value of H0 = 73.2 ± 1.3 is W derived combing anchor distances from the MW, NGC 4258 and LMC. Figure 3. H0 as a function of RW, as used in the Wesenheit Using the same data with RW = 0.386, using double calibration with respect to RW (V − I), in SNIa hosts and P-L relations, doing a full count-rate non-linearity cor- anchor galaxies. rection following Riess et al. (2019), identifying [O/H] = [Fe/H] using Z⊙ = 8.824, and fitting for the residual GAIA offset zp simultaneously with all other parame- 5.2. Fitting for RW ters, we obtain H0 = 73.1 ± 1.3. Here, we have added a The fact that dust extinction extrapolated to the H- W scatter in the P-L relation of σ(mH )=0.0682, to give band is very uncertain naturally opens up for the option 2 a χ /dof = 1. Despite slight differences in the analysis of fitting for the value of RW, either a global value com- method, this value is in very good agreement with the mon for all galaxies or individually for all galaxies. The value in Riess et al. (2021), and in 4.1 σ tension with the first option gives RW =0.31±0.02 with H0 = 73.7±1.3 Planck value of H0 = 67.4 ± 0.5 (Aghanim et al. 2020). with Planck tension 4.4 σ. If we allow for RW to vary between galaxies we obtain Systematic Shift of 5.1. RW the result in Figure 4, with host galaxies systematically The inferred value of H0 will shift if there is a system- favoring lower values for RW, and H0 = 66.9 ± 2.5 in atic offset in RW between anchor and SNIa host galaxies. agreement with the Planck value. With a prior con- We derive values for H0 when varying the value of RW straint on the individual values of RW, centered on in the anchor(s) and the SNIa hosts, see Figure 3. For a RW = 0.386, the inferred H0 will transition smoothly common RW (indicated by the dotted line), the inferred from H0 = 66.7 ± 2.7 (for an infinite prior width) to value of H0 decreases when RW is increased. Also, H0 H0 = 73.1 ± 1.3 (for zero prior width). Imposing weak is decreased when RW is larger in anchor than in host prior constraints of RW = 0.45 ± 0.35, guided by the galaxies. Given the result in Figure 3, one could argue discussion in Section 2, we obtain H0 = 69.4 ± 2.6 with that a simple solution to the Hubble tension, would be a Planck tension < 1 σ. systematic shift of RW between anchor and host galax- 5.3. Individual P-L relations ies of ∆RW ∼ 0.15, bringing H0 as inferred from SNIa The Hubble Tension Bites the Dust 7

0.8 Anchor -1.5 Anchor Host Host R = 0.386 b = -3.3 0.6 W W -2.0 0.4 W

W -2.5 b R 0.2 -3.0 0.0 -3.5 -0.2 MWLMCN4258M31M101N1015N1309N1365N1448N2442N3021N3370N3447N3972N3982N4038N4424N4536N4639N5584N5917N7250U9391 MWLMCN4258M31M101N1015N1309N1365N1448N2442N3021N3370N3447N3972N3982N4038N4424N4536N4639N5584N5917N7250U9391

Figure 4. Fitting individual luminosity-color relations Figure 5. Fitting individual P-L relations bW to Cepheid RW (V − I) to Cepheid data. Anchor galaxies are denoted in data. Anchor galaxies are denoted in brown and SNIa host brown and SNIa host galaxies in petrol. The dotted line cor- galaxies in petrol. The dotted line corresponds to bW = responds to RW = 0.386 as commonly assumed for Cepheid −3.26, indicating the value obtained assuming a global P-L calibration. relation.

So far, we have assumed that all Cepheids can be de- Monte Carlo samples. The mean color excess range s ˆ ˆ scribed by a global P-L relation, described by bW and from E(V − I) = 0.28 in the LMC to E(V − I) = 0.69 l bW. However, since there is evidence that the P-L re- in the MW, see also Figure 10. Calibrating using lation can vary between galaxies (Tammann et al. 2011; RE Eˆ(V − I), for a fixed value of RE =0.386, we obtain Efstathiou 2020), in a similar spirit to our approach of H0 = 73.0 ± 1.3, showing the insensitivity of calibration allowing RW to vary between galaxies, we investigate method for fix values of RW and RE. to what extent relaxing this assumption will affect the The derived value of H0 depends on the assumed value inferred Hubble constant. We will allow for individual for RE in the anchor(s) and the SNIa hosts as depicted galactic values of bW,i to be fitted for simultaneously in Figure 6. We note a weaker dependence on the values with all other parameters, in this case restricting to the of RE when compared to the sensitivity of H0 to RW in same P-L relation for short and long period Cepheids. the Wesenheit calibration, as shown in Figure 3. For a fixed global value of RW = 0.386, the resulting bW,i are shown in Figure 5. The fact that the fitted 6.1. Fitting for RE b are systematically higher in SNIa host galaxies com- W,i Fitting for a global value of RE, common for all galax- pared to hosts, the inferred Hubble constant is increased ies, gives RE =0.31 ± 0.02 with H0 = 73.6 ± 1.3 increas- from H0 = 73.1 ± 1.3 to H0 = 76.5 ± 1.9. If we allow ing the Planck tension to 4.4 σ. for both individual P-L and C-L relations, the result is We next allow for RE to vary between galaxies. Since again shifted down to H = 66.1 ± 2.7. 0 RE represents a combination of corrections to both dust and intrinsic colors, both with large uncertainties in 6. RESULTS FOR COLOR EXCESS CALIBRATION the NIR, we only impose weak prior constraints on RE Eˆ(V − I) their values RE = 0.45 ± 0.35 motivated by observa- We next compare with results derived when color cal- tions presented in Section 2, obtaining H0 = 71.8 ± 1.6 ibrating the Cepheid sample with respect to the esti- km/s/Mpc, in 2.7 σ tension with the Planck value, see mated color excess. We derive Eˆ (V − I) subtracting Figure 7. The priors keep the 1σ upper limit of the fitted mean intrinsic colors as estimated in Tammann et al. values well within RE > 0. Comparing with the results (2011), including the quoted uncertainties and a 0.075 for RW in Figure 4, it is evident that we get very similar dispersion in the mean intrinsic Cepheid color between results for RE with increased error bars. The one ex- galaxies (inferred from the difference between LMC and ception is LMC for which the preferred value for RE is MW Cepheids). These colors are in good agreement significantly larger than for RW. Not imposing any pri- with results in Pejcha & Kochanek (2012). The intrin- ors on RE yields H0 = 70.9 ± 1.7 km/s/Mpc, in which sic color uncertainties are included generating random case for N3370 only the 2σ upper limit has RE > 0. 8 Mortsell¨ et al.

1.0 84 R W CMB R SH0ES 66 RE 81 E (no prior) 0.8 79 68

76 0.6 70

74 (anchors) E 0.4 72 R 71

69 0.2 74 65 70 75 H 66 0 76 82 78 80 64 0.0 Figure 8. Results for H0 for the two main analyses em- 0.0 0.5 1.0 1.5 ployed in this paper. The solid black line is for individ- R E (hosts) ual galactic values of RW using Wesenheit magnitudes. The dark petrol lines are for individual RE using color excesses, with the solid line assuming prior values RE = 0.45 ± 0.35. Figure 6. H0 as a function of RE in the SNIa hosts and the anchor galaxies when color calibrating Cepheids with respect The dashed petrol line is ﬁtted using the Wesenheit calibration with RW = 0.386 as in Riess et al. (2021) and the to the estimated color excess RE Eˆ(V − I). dashed brown region indicates the 1 σ region from Planck (Aghanim et al. 2020). Allowing also for individual galactic P-L relations, we obtain H0 = 73.5 ± 2.0 (3.0 σ tension), assuming priors We have argued that the value inferred for the Hub- on R . E ble constant from SNIa distance measurements is quite sensitive to the choice of Cepheid calibration method. 0.8 Speciﬁcally, we have compared results when color calibrating the Cepheid magnitudes with respect to ob- 0.6 served colors and estimated color excesses. In both cases, guided by the lack of evidence for global dust 0.4 extinction properties at NIR wavelengths, we have al- lowed for individual color calibrations in each galaxy. E

R 0.2 Regardless of whether we calibrate with respect to observed colors or estimated color excesses, the correction terms RW (V − I) and RE Eˆ (V − I) only have clear in- 0.0 VI VI terpretations if {RW, RE} = RH = βH , correcting for Anchor Host both dust extinction and intrinsic color variations. Both R = 0.386 -0.2 E approaches could thus be regarded as partly phenomeno-

MWLMCN4258M31M101N1015N1309N1365N1448N2442N3021N3370N3447N3972N3982N4038N4424N4536N4639N5584N5917N7250U9391 logical, and partly as a physical model parameterization. For the color excess calibration, we derive H0 = 71.8 ± 1.6 in 2.7 σ tension with the value inferred from Planck when using a weak prior of RE = 0.45 ± 0.35. Figure 7. The result of ﬁtting individual galactic values for With no prior, H0 = 70.9±1.7 (2.0 σ tension). Calibrat- RE using the color excess Cepheid calibration, imposing prior ing with respect to observed colors yields H0 = 66.9±2.5 constraints RE = 0.45 ± 0.35.Anchor galaxies are denoted in in agreement with the Planck value, see Figure 8. These brown and SNIa host galaxies in petrol. The dotted line corresponds to RE = 0.386. results are stable to the possibility of also allowing for individual galactic P-L relations. Results for individual anchors are presented in Section C. The lower H0 compared to the case of a ﬁxed global color calibration is 7. SUMMARY AND DISCUSSION driven by the preferred lower values of RW and RE in The Hubble Tension Bites the Dust 9

6 Anchor 5 Host SNIa R = 3.1 4 V

BV V 3 R 2 1 0 MW LMC N4258M31 M101N1015 (2011fe)N1309 (2009ig)N1365 (2002fk)N1448 (2012fr)N2442 (2001el)N3021·(1995al) (2015F)N3370N3447 (1994ae)N3972 (2012ht)N3982 (2011by)N4038 (1998aq)N4424 (2007sr)N4536 (2012cg)N4639 (1981B)N5584 (1990N)N5917 (2007af)N7250 (2005cf)U9391 (2013dy) (2003du)

BV Figure 9. Comparing results for RV as ﬁtted using the color excess Cepheid calibration (dark petrol) and independently from BV SNIa colors in Burns et al. (2018) (light petrol). The dotted line corresponds to RV = 3.1. Shown to the right of the plot are BV the total probability distribution functions for RV for the SNIa hosts, as derived from Cepheid and SNIa colors, respectively.

SNIa hosts. Comparing the SNIa host dust properties yields H0 = 57.3 ± 2.7, in 3.7 σ tension with Planck. VI as inferred here from Cepheid data using color excesses With the color excess calibration, for RE,I = 1.27, we with independent constraints from the SNIa color data obtain H0 = 69.6 ± 1.5, in 1.4 σ tension with the CMB from Burns et al. (2018) (derived using the CCM pa- value, whereas allowing for individual galactic values of VI rameterization in Cardelli et al. 1989) shows a reassur- RE,I yields H0 = 66.9 ± 1.3, in agreement with Planck. ing agreement, see Figure 9, with the one exception of SNIa 2011fe in M101. Results for different calibration choices are summa- 1.0 rized in Table 1 in Section B. From the quality of the fits, there is no clear preference for any of the calibration methods. Regardless of calibration method, since Cepheid col- 0.5 ors and periods are correlated (Tammann et al. 2011), so will the inferred P-L and C-L relations, i.e., the pa- [P] MW rameters bW and RW or RE. This may be of importance 0.0 LMC if there are color selection effects related to the fact that N4258 longer period Cepheids are brighter, see Figure 10. We M31 Host have tested the possible impact of such an effect by im- -0.5 posing cuts on the observed color V−I. In Figure 11, we show the fitted H0 as a function of the cut in (V − I) we 0.5 1.0 1.5 2.0 2.5 apply, when calibrating using color excesses for a fixed V-I RE = 0.386. The trend of obtaining a lower H0 when cutting out redder Cepheid is a common feature for all Figure 10. Observed Cepheid colors V−I versus the periods calibration methods we have tested. [P ] = log P − 1. The dotted lines correspond to the upper Finally, we note that when estimating distances using and lower limit of the mean intrinsic MW Cepheid color as the I-band for which color corrections are larger, unsur- estimated in Tammann et al. (2011). prisingly, the sensitivity of the result to the choice of calibration method is even larger. For example, with With the limited information at hand regarding dust the Wesenheit calibration, assuming a fixed value of extinction and intrinsic C-L relations for Cepheids at RVI =1.27 (corresponding to RVI =0.386), we obtain NIR wavelengths, and our current inability to distin- W,I H guish between them in most cases, the color calibration H0 = 70.4 ± 1.5, in 1.9 σ tension with the CMB value, whereas allowing for individual galactic values of RVI of Cepheid magnitudes introduces large uncertainties in W,I the local distance ladder. Neither approach employed in 10 Mortsell¨ et al.

this paper is in one-to-one correspondence with an un- 80 derlying physical model and there is no clear evidence CMB in the data for any of them. Given that results dif- SH0ES fer by δH0 ∼ 5, similar to the diﬀerence between the 75 SH0Es and the Planck value and exceeding individual statistical errors by a factor & 2, we conclude that sys-

0 tematic uncertainties related to the choice of Cepheid

H 70 color-luminosity calibration method have the potential to resolve the current Hubble tension. 65 1 EM acknowledges support from the Swedish Research 60 2 Council under Dnr VR 2020-03384. AG acknowledges 3 support from the Swedish Research Council under Dnr 0.8 1.0 1.2 1.4 1.6 1.8 2.0 4 VR 2020-03444, and the Swedish National Space Board, (V-I) max 5 grant 110-18.

Figure 11. Fitted H0 as a function of the cut in (V − I) for a ﬁxed value of RE = 0.386 when calibrating using estimated color excesses.

APPENDIX

A. SYSTEM OF EQUATIONS Following Riess et al. (2016), we collect all data points and their corresponding uncertainties and possible correlations W in the matrices Y and C. This includes the Wesenheit magnitudes of all Cepheids, mH,i,j, including the anchors. The W 5 exception is the Milky Way Cepheids for which we use mπ,j ≡ mH,j − 10 + ln 10 ln π. Next, we have the measured anchor distances, µN4258,µLMC and possibly µM31. Finally, data points include the B-band SNIa magnitudes in the Cepheid hosts, mB,i:

mπ,j  mW  H,1,j σ2(m ) 0 ... 0 . π,j  .  .  .   2 W .    0 σ (mH ) 0 ... .  mW   H,19,j   . ..   W   . .  mH,N4258,j       σ2(µ )  Y =  mW  , C =  N4258  (A1)  H,LMC,j   2     σ (µLMC)   µN4258       σ2(m )   µ   B,1   LMC   .     ..   mB,1     .   2   .   0 σ (mB,19)  .       m   B,19  The Hubble Tension Bites the Dust 11

Collecting the model parameters in the matrix X

µ0,1  .  .    µ0,19    µ   N4258    µLMC    X =  mW  , (A2)  H   s   bW     bl   W     ZW     zp     M   B  we can relate data and parameters through Y = AX where in schematic form

−1 s l −5π1 0 ... 1 [P]MW,1 [P]MW,1 [M/H]1 ln 10 0 .  . −1  −5π   s l N  0 1 [P]MW,N [P]MW,N [M/H]N ln 10 0  .  s l . 1 0 ... 1 [P] [P] [M/H] 0 .   . . . . .  . 1 0 ... 1 . . . .  A =   . (A3)  ... 1 0 0       1 0    1 0 ... 1     01 0 ... 1 . . . . . .   0 ... 1 0 ... 1   Note that the ﬁrst N rows correspond to equation 15 and so forth, so that A is a nparam × ndata matrix. We can solve for the parameter matrix X and its covariance matrix Σ analytically −1 Σ = AT C−1A , X = Σ AT C−1Y . (A4) Note that the formalism can easily extended to ﬁt also for, e.g., RW and RE, and that parameter priors are incorporated as additional data points.

B. MODEL SELECTION

The choice of C-L and P-L calibration method can have a large impact on the inferred H0. Unfortunately, from data alone, it is not obvious which of calibration models is preferred. Specifically, there is no clear preference in data for color calibrating with respect to observed colors or estimated color excesses. As expected, allowing for more freedom in the model, the fit will improve2. In terms of model selection, results are ambiguous, see Table 1. Here, the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) are defined by

2 AIC ≡ 2npar + χ , 2 BIC ≡ npar ln ndat + χ , (B5)

2 When comparing the quality of the ﬁt for diﬀerent models, we add the same additional scatter to the P-L relation. When constraining parameters within a given model, the scatter is adjusted to give a χ2/dof = 1. 12 Mortsell¨ et al.

Table 1. Resulting H0 and quality of ﬁt for diﬀerent color calibration models for Wesenheit calibration, RW (V − I), and color excess calibration, RE Eˆ (V − I). The prior for the color excess calibration refer to RE = 0.45 ± 0.35. 2 Wesenheit cal. H0 (Planck tension) χmin AIC BIC RW = 0.386 73.1 ± 1.3 (4.1 σ) 1616.6 1672.6 1823.9 Global RW 73.7 ± 1.3 (4.4 σ) 1602.4 1660.4 1817.1 Individual RW 66.9 ± 2.5 (0.2 σ) 1549.2 1651.2 1926.9 Individual bW 76.5 ± 1.9 (4.5 σ) 1566.0 1664.0 1928.9 Individual RW and bW 66.4 ± 2.8 (0.4 σ) 1497.4 1641.4 2030.6 2 Color excess cal. H0 (Planck tension) χmin AIC BIC RE = 0.386 73.0 ± 1.3 (4.0 σ) 1612.7 1668.7 1820.0 Global RE 73.6 ± 1.3 (4.4 σ) 1598.9 1656.9 1813.7 Individual RE (w prior) 71.8 ± 1.6 (2.7 σ) 1610.2 1712.2 1988.6 Individual RE (wo prior) 70.9 ± 1.7 (2.0 σ) 1533.9 1635.9 1911.6 Individual bW 76.1 ± 1.9 (4.4 σ) 1563.5 1661.5 1926.3 Individual RE and bW (w prior) 73.5 ± 2.0 (3.0 σ) 1504.8 1648.8 2039.0 Individual RE and bW (wo prior) 72.4 ± 1.9 (2.5 σ) 1487.0 1631.0 2020.2

with the latter more penalizing for models with extra parameters for ndat > 8. In terms of the AIC, fitting for individual values of RW or RE and bW is the preferred model, whereas in terms of BIC, a global fitted value yields the best result. As it stands, it is unclear to what degree these results provide guidance on the choice of calibration model. C. RESULTS FOR INDIVIDUAL ANCHORS In Figure 12, we show the inferred Hubble constant from each individual anchor. Here, we have used individually fitted RW and RE for each galaxy but assumed a global P-L relation. Also allowing for bW to vary between galaxies yields similar results. Dotted lines are for a global value RW =0.386 using the Wesenheit calibration, closest resembling the case in Riess et al. (2021). The solid lines are for the case of fitting for individual RW and RE. The main difference in both cases, besides broadening the likelihoods, is the shift of LMC anchor result.

N4258 N4258 MW MW LMC LMC CMB CMB

50 55 60 65 70 75 80 85 65 70 75 80 H H 0 0

Figure 12. Results for H0 for different anchor distances. Dotted lines are fitted using the Wesenheit calibration with RW = 0.386 as in Riess et al. (2021). Left panel: Solid lines are fitted for individual galactic values of RW using Wesenheit magnitudes. Right panel: Solid lines are fitted for individual galactic values of RE using estimated color excesses.

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